scholarly journals A guaranteed deterministic approach to superhedging: mixed strategies and game equilibrium

2020 ◽  
Vol 12 (1) ◽  
pp. 60-90
Author(s):  
Сергей Николевич Смирнов ◽  
Sergey Smirnov
Keyword(s):  
Transaction Costs ◽  
A Priori ◽  
Contingent Claim ◽  
Mixed Strategies ◽  
Deterministic Approach ◽  
Game Equilibrium ◽  
Game Theoretic ◽  
Price History ◽  
Pure Strategies ◽  
Trading Constraints

For a discrete-time superreplication problem, a guaranteed deterministic formulation is considered: the problem is to ensure a cheapest coverage of the contingent claim on an option under all scenarios which are set using a priori defined compacts, depending on the price history: price increments at each moment of time must lie in the corresponding compacts. The market is considered with trading constraints and without transaction costs. The statement of the problem is game-theoretic in nature and leads directly to the Bellman - Isaacs equations. In this article, we introduce a mixed extension of the ``market'' pure strategies. Several results concerning game equilibrium are obtained.

scholarly journals A Guaranteed Deterministic Approach to Superhedging—The Case of Convex Payoff Functions on Options

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1246 ◽  
Author(s):  
Sergey Smirnov
Keyword(s):  
A Priori ◽  
Contingent Claim ◽  
Mixed Strategies ◽  
Deterministic Approach ◽  
Deterministic Problem ◽  
History Of ◽  
Game Theoretic ◽  
Payoff Functions ◽  
Special Case ◽  
Problem Setting

This paper considers super-replication in a guaranteed deterministic problem setting with discrete time. The aim of hedging a contingent claim is to ensure the coverage of possible payoffs under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts that depend on the history of prices. The increments of the price at each moment in time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game–theoretic interpretation of pricing American options implies that the corresponding Bellman–Isaacs equations hold for both pure and mixed strategies. In the present paper, we study some properties of the least favorable (for the “hedger”) mixed strategies of the “market” and of their supports in the special case of convex payoff functions.

scholarly journals Guaranteed Deterministic Approach to Superhedging: Case of Binary European Option

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Sergey N. Smirnov ◽  
Andrey Yu. Zanochkin
Keyword(s):  
A Priori ◽  
Payoff Function ◽  
Market Model ◽  
Solution Properties ◽  
Deterministic Approach ◽  
European Option ◽  
Bellman Equations ◽  
Contingent Liability ◽  
Game Theoretic ◽  
Trading Constraints

For the superreplication problem with discrete time, a guaranteed deterministic formulation is considered: the problem is to guarantee coverage of the contingent liability on sold option under all admissible scenarios. These scenarios are defined by means of a priori defined compacts dependent on price prehistory: the price increments at each point in time must lie in the corresponding compacts. In a general case, we consider a market with trading constraints and assume the absence of transaction costs. The formulation of the problem is game theoretic and leads to the Bellman–Isaacs equations. This paper analyses the solution to these equations for a specific pricing problem, i.e., for a binary option of the European type, within a multiplicative market model, with no trading constraints. A number of solution properties and an algorithm for the numerical solution of the Bellman equations are derived. The interest in this problem, from a mathematical prospective, is related to the discontinuity of the option payoff function.

scholarly journals A guaranteed deterministic approach to superhedging: most unfavorable scenarios of market behaviour and moment problem

2020 ◽  
Vol 12 (3) ◽  
pp. 50-88
Author(s):  
Сергей Николаевич Смирнов ◽  
Sergey Sergey
Keyword(s):  
Bellman Equation ◽  
A Priori ◽  
Theoretical Interpretation ◽  
Mixed Strategies ◽  
Deterministic Approach ◽  
Step Method ◽  
Risky Assets ◽  
Deterministic Problem ◽  
Game Equilibrium ◽  
Problem Setting

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure the coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game-theoretical interpretation implies that the corresponding Bellman-Isaac equations hold, both for pure and mixed strategies. In the present paper, we propose a two-step method of solving the Bellman equation arising in the case of (game) equilibrium. In particular, the most unfavorable strategies of the `market can be found in the class of the distributions concentrated at most in n+1 point, where n is the number of risky assets.

scholarly journals A Game Theoretic Approach for Eco-Design and Remanufacturing Considering Take-Back Policy

2020 ◽  
Vol 12 (17) ◽  
pp. 7174
Author(s):  
Xiaoxiao Chang ◽  
Guangye Xu ◽  
Qian Wang ◽  
Yongguang Zhong
Keyword(s):  
Environmental Impact ◽  
Stackelberg Game ◽  
Theoretic Approach ◽  
Subsidy Policy ◽  
Monetary Value ◽  
Game Equilibrium ◽  
Take Back ◽  
Game Theoretic ◽  
The Government ◽  
Game Theoretic Approach

This paper mainly aims at investigating the governments’ take-back policy of penalty or subsidy that motivates eco-design or remanufacturing. For this purpose, we consider a two-stage Stackelberg game between a government and a manufacturer. The government first decides to impose a take-back penalty or offer a take-back subsidy, and then the manufacturer selects to remanufacture or invest in eco-design as a response to the take-back policy. Upon analyzing and comparing game equilibrium, we find that the government prefers to offer a subsidy policy for eco-design and to impose a penalty policy for remanufacturing. The manufacturer will decide on investing in eco-design when the monetary value of the environmental impact of landfill and eco-design coefficient is medium. However, if the eco-design coefficient is high, the manufacturer practices remanufacturing instead of eco-design whether penalized and subsidized. The present study provides a set of guidelines in practical managerial recommendations for governments and manufacturers.

Advertising Rivalry in the U. S. Automobile Industry: A Test of Bain's Hypothesis

1994 ◽  
Vol 38 (2) ◽  
pp. 40-51 ◽  
Author(s):  
Lall B. Ramrattan
Keyword(s):  
Automobile Industry ◽  
Serial Correlation ◽  
Product Characteristic ◽  
Theoretic Model ◽  
Mixed Strategies ◽  
Solution Concepts ◽  
Network Television ◽  
Price Collusion ◽  
Game Theoretic ◽  
Game Theoretic Model

This paper examines Bain's Hypothesis that firms in the automobile industry engage in advertising competition and price collusion. It develops a game theoretic model, basing price on product characteristic and advertising on pure and mixed strategies. Solution concepts such as Cournot, Nash and Prisoner's dilemma are possible. The paper then moves into regression results of advertising outlays in newspaper, general magazines, spot television, and network television, given the firm's cashflow, OPEC influence, and Leader-follower hypotheses. When multicolinearity and serial correlation are adjusted for, the results corroborate Bain's advertising hypothesis with price collusion.

On the Foundations of Nash Equilibrium

1996 ◽  
Vol 12 (1) ◽  
pp. 67-88 ◽  
Author(s):  
Hans Jørgen Jacobsen
Keyword(s):  
Game Theory ◽  
Nash Equilibrium ◽  
Cooperative Game Theory ◽  
Mixed Strategy ◽  
Pure Strategy ◽  
Mixed Strategies ◽  
Positive Theory ◽  
Equilibrium Concept ◽  
Strategy Equilibrium ◽  
Pure Strategies

The most important analytical tool in non-cooperative game theory is the concept of a Nash equilibrium, which is a collection of possibly mixed strategies, one for each player, with the property that each player's strategy is a best reply to the strategies of the other players. If we do not go into normative game theory, which concerns itself with the recommendation of strategies, and focus instead entirely on the positive theory of prediction, two alternative interpretations of the Nash equilibrium concept are predominantly available.In the more traditional one, a Nash equilibrium is a prediction of actual play. A game may not have a Nash equilibrium in pure strategies, and a mixed strategy equilibrium may be difficult to incorporate into this interpretation if it involves the idea of actual randomization over equally good pure strategies. In another interpretation originating from Harsanyi (1973a), see also Rubinstein (1991), and Aumann and Brandenburger (1991), a Nash equilibrium is a ‘consistent’ collection of probabilistic expectations, conjectures, on the players. It is consistent in the sense that for each player each pure strategy, which has positive probability according to the conjecture about that player, is indeed a best reply to the conjectures about others.

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